Can a Vector Be Expanded into a Field in Physics?

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A vector can indeed be expanded into a field in physics, as demonstrated by the relationship between gravitational potential and force. The equation F = -∇V illustrates that the vector representing force is derived from the gradient of the scalar potential field V. This means that a vector, when defined across a plane, can create a vector field that describes how the force varies in space. Therefore, expanding a vector throughout a plane effectively results in a field representation. Understanding this concept is crucial for analyzing physical phenomena involving forces and potentials.
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Hey everyone,
I was just wondering can a vector be expanded into a field? I.e. take a gravitational potential that can be described by a vector (F = -\nablaV). Can this vector then be expanded into a field given a plane with the point of attraction at the origin, for simplicity.

Thanks
 
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Vectors by themselves are just something that has both magnitude and direction. But what you described (F = - del(V)) is a vector field, which... yeah, is a field.
 
So if I take a vector and expand it throughout some arbitrary plane it makes a field?
 

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